Chapter Five

DIFFUSION ANDRESISTIVITY

DIFFUSION AND MOBILITY IN WEAKLY IONIZED GASES 5.1

The infinite, homogeneous plasmas assumed in the previous chapter forthe equilibrium conditions are, of course, highly idealized. Any realisticplasma will have a density gradient, and the plasma will tend to diffusecoward regions of low density. The central problem in controlled ther-monuclear reactions is to impede the rate of diffusion by using a magneticfield. Before tackling the magnetic field problem, however, we shallconsider the case of diffusion in the absence of magnetic fields. A furthersimplification results if we assume that the plasma is weakly ionized, sothat charge particles collide primarily with neutral atoms rather thanwith one another. The case of a fully ionized plasma is deferred to alater section, since it results in a nonlinear equation for which there areI’ew simple illustrative solutions. In any case, partially ionized gases arenot rare: High-pressure arcs and ionospheric plasmas fall into thiscategory, and most of the early work on gas discharges involved fractionalionizations between lo-’ and 10e6, when collisions with neutral atomsare dominant.

The picture, then, is of a nonuniform distribution of ions andelectrons in a dense background of neutrals (Fig. 5-1). As the plasmaspreads out as a result of pressure-gradient and electric field forces, theindividual particles undergo a random walk, colliding fre’quently withthe neutral atoms. We begin with a brief review of definitions fromatomic theory. 155

h

156ChapterFive

FIGURE 5-1

O00 oovo

OO

IO0 0 O” O 0

O 0 od:

???? ???

??

? ????

???????? ? ??

Diffusion of gas atoms by randomcollisions.

5.1.1 Collision Parameters

When an electron, say, collides with a neutral atom, it may lose anyfraction of its initial momentum, depending on the angle at which it,rebounds. In a head-on collision with a heavy atom, the electron. canlose twice its initial momentum, since its velocity reverses sign after thecollision. The probability of momentum loss can be expressed in termsof the equivalent cross section u that the atoms would have if they wereperfect absorbers of momentum.

In Fig. 5-2, electrons are incident upon a slab of area A and thicknessdx containing n, neutral atoms per m3. The atoms are imagined to beopaque spheres of cross-sectional area a; that is, every time an electron

FIGURE 5.2 Illustration of the definition of cross section.

comes within the area blocked by the atom, the electron loses all of itsmomentum. The number of atoms in the slab is

n,A dx

The fraction of the slab blocked by atoms is

n,&dx/A = n,udx

If a flux r of electrons is incident on the slab, the flux emerging on theother side is

I?’ = r(l - n,adx)

Thus the change of r with distance is

dI’/dx = -n,or

or

r=roe --%m G r. e-xl*m 15-11

In a distance λm, the flux would be decreased to 1/e of its initial value.The quantity A, is the mean free path for collisions:

pJq 15-21

After traveling a distance λm, a particle will have had a good probabilityof making a collision. The mean time between collisions, for particles ofvelocity v, is given by

7 = Am/v

and the mean frequency of collisions is-1

7 = v/A,,, = n,crv [5- 3]

If we now average over particles of all velocities v in a Maxwelliandistribution, we have what is generally called the collision frequency V:

v = n,uv 15-41

Diffusion Parameters 5.1.2

The fluid equation of motion including collisions is, for any species,

157Diffusion and

Resistivity

1 =fenE-Vfi-mnvv 15-51

158 where again the f indicates the sign of the charge. The averagingChapterFive

process used to compute v is such as to make Eq. [5-5] correct; we neednot be concerned with the details of this computation. The quantity vmust, however, be assumed to be a constant in order for Eq. [5-5] to beuseful. We shall consider a steady state in which h/at = 0. If v issufficiently small (or v sufficiently large), a fluid element will not moveinto regions of different E and Vp in a collision time, and the convectivederivative dv/dt will also vanish. Setting the left-hand side of Eq. [5-5]to zero, we have, for an isothermal plasma,

v=&enE-KTVn)rnnv

1541KT Vn=&E--w-

The coefficients above are called the mobility and the diffusion coefficient :

r5-71

Diffusion coefficient 15-81

These will be different for each species. Note that D is measured inm’/sec. The transport coefficients e and D are connected by the Einsteinrelation:

F-91

With the help of these definitions, the flux ri of the jth species can bewritten

15-101

Fick’s law of diffusion is a special case of this, occurring when eitherE = 0 or the particles are uncharged, so that p = 0:

pjFick’s law 15.111

This equation merely expresses the fact that diffusion is a random-walkprocess, in which a net flux from dense regions to less dense regionsI occurs simply because more particles start in the dense region. This flux

is obviously proportional to the gradient of the density. In plasmas, Fick’slaw is not necessarily obeyed. Because of the possibility of organizedmotions (plasma waves), a plasma may spread out in a manner which is not truly random.

159Diffusion and

Resistivity

DECAY OF A PLASMA BY DIFFUSION 5.2

Ambipolar Diffusion 5.2.1

We now consider how a plasma created in a container decays by diffusionto the walls. Once ions and electrons reach the wall, they recombinethere. The density near the wall, therefore, is essentially zero. The fluidequations of motion and continuity govern the plasma behavior; but ifthe decay is slow, we need only keep the time derivative in the continuityequation. The time derivative in the equation of motion, Eq. [5-5] willbe negligible if the collision frequency v is large. We thus have

anz+V’rj=O [5- 121

with rj given by Eq. [5-10]. It is clear that if I’i and r, were not equal,a serious charge imbalance would soon arise. If the plasma is muchlarger than a Debye length, it must be quasineutral; and one wouldexpect that the rates of diffusion of ions and electrons would somehowadjust themselves so that the two species leave at the same rate. Howthis happens is easy to see. The electrons, being lighter, have higherthermal velocities and tend to leave the plasma first. A positive chargeis left behind, and an electric field is set up of such a polarity as to retardthe loss of electrons and accelerate the loss of ions. The required E fieldis found by setting Ii = I, = Γ. From Eq. [5-10], we can write

[5- 131

[5- 141

160ChapterFive

The common flux Γ is then given by

r = piDi -0,

µi

- +I&

Vn -DiVn

= _ PiDc + PcDi V nµi +

- I&

[5- 151

This is Fick’s law with a new diffusion coefficient

called the ambipolar diffusion coefficient. If this is constant, Eq. [5-12]becomes simply

an/at = Da V2n [5-171

The magnitude of Da can be estimated if we take cc, >> pi. That thisis true can be seen from Eq. [5-7]. Since v is proportional to the thermalvelocity, which is proportional to m-l/2, p is proportional to m-1’2.Equations [5-16] and [5-9] then give

0, zzDi +&De =Di ++ELe

[5- 181t

For IY, = Ti, we have

D, a 2Di p- 191

The effect of the ambipolar electric field is to enhance the diffusion ofions by a factor of two, but the diffusion rate of the two species togetheris primarily controlled by the slower species.

Diffusion in a Slab

The diffusion equation [5-17] can easily be solved by the method ofseparation of variables. We let

n(r,t) = T(t)S(r) [5-201

whereupon Eq. [5-17], with the subscript on 0, understood, becomes

SdT=DTV2Sdt

[5-411

IdT372~T dt S

[5- 221

Since the left side is a function of time alone and the right side a functionof space alone, they must both be equal to the same constant, which weshall call - 1/τ. The function T then obeys the equation

dT Tz= -G-

with the solutionT = Toe-r’r [5-241

The spatial part S obeys the equation

[5-!&s]

v”s= -&sIn slab geometry, this becomes

with the solution

S =A cos*+B sin,&

[5-251

15-261

[5-271

We would expect the density to be nearly zero at the walls (Fig. 5-3) andto have one or more peaks in between. The simplest solution is that witha single maximum. By symmetry, we can reject the odd (sine) term inEq. [5-27]. The boundary conditions S = 0 at x = *L then requires

or

Combining Eqs. [5-20] [5-24], [5-27], and [5-28], we have

-t/7 m-xn =nOe

cosqL

[5-281

[5-291

161Diffusion and

Resistivity

162ChapterFive

FIGURE 5-3 Density of a plasma at varioustimes as it decays by diffusionto the walls.

This is called the lowest diffusion mode. The density distribution is a cosine,and the peak density decays exponentially with time. The time constantT increases with L and varies inversely with D, as one would expect.

There are, of course, higher diffusion modes with more than onepeak. Suppose the initial density distribution is as shown by the top curvein Fig. 5-4. Such an arbitrary distribution can be expanded in a Fourierseries:

((1 + $Tx m7rx

n =no Caicos +Cbm sin-1 L nz L >

15-301

We have chosen the indices so that the boundary condition at x = AL isautomatically satisfied. To treat the time dependence, we can try asolution of the form

n =no(Gale -L/7, cos

(I + i)nx+ C 6, e+“m sin y

L m >[5-311

1

Substituting this into the diffusion equation [5-17], we see that eachcosine term yields a relation of the form

[5-341

163Diffusion and

Resistivity

Decay of an initially nonuniform FIGURE 5-4plasma, showing the rapid disappear-ance of the higher-order diffusionmodes.

and similarly for the sine terms. Thus the decay time constant for theI th mode is given by

L1 ]21T1 = (1 + $)T 5

[5-331

The fine-grained structure of the density distribution, corresponding tolarge 1 numbers, decays faster, with a smaller time constant ~1. Theplasma decay will proceed as indicated in Fig. 5-4. First, the fine structurewill be washed out by diffusion. Then the lowest diffusion mode, thesimple cosine distribution of Fig. 5-3, will be reached. Finally, the peakdensity continues to decay while the plasma density profile retains thesame shape.

Diffusion in a Cylinder 5.2.3

The spatial part of the diffusion equation, Eq. [5-25], reads, in cylindricalgeometry,

d2S 1 dS 1dr2+--+rdr zs=’

[5- 341

This differs from Eq. [5-26] by the addition of the middle term, whichmerely accounts for the change in coordinates. The need for the extraterm is illustrated simply in Fig. 5-5. If a slice of plasma in (A) is movedtoward larger x without being allowed to expand, the density would

164ChapterFive

X

A B

FIGURE 5-5 Motion of a plasma slab in rectilinear and cylindrical geometry, illustrating thedifference between a cosine and a Bessel function.

remain constant. On the other hand, if a shell of plasma in (B) is movedtoward larger r with the shell thickness kept constant, the density wouldnecessarily decrease as 1/r. Consequently, one would expect the solutionto Eq. [5-34] to be like a damped cosine (Fig. 5-6). This function is calleda Bessel function of order zero, and Eq. [5-34] is called Bessel's equation (of order zero). Instead of the symbol cos, it is given the symbol Jo. Thefunction Jo(r/[ll~]“~) is a solution to Eq. [5-34], just as cos [x/(&)“~] isa solution to Eq. [5-26]. Both cos kx and J&r) are expressible in terms

12 14 kr

FIGURE 5-6 The Bessel function of order zero.

of infinite series and may be found in mathematical tables. Unfortunately,Bessel functions are not yet found in hand calculators.

To satisfy the boundary condition n = 0 at r = a, we must set(,/(DT)“* equal to the first zero of Jo; namely, 2.4. This yields the decaytime constant 7. The plasma again decays exponentially, since the tem-poral part of the diffusion equation, Eq. [5-23], is unchanged. We havedescribed the lowest diffusion mode in a cylinder. Higher diffusionmodes, with more than one maximum in the cylinder, will be given interms of Bessel functions of higher order, in direct analogy to the caseof slab geometry.

STEADY STATE SOLUTIONS

In many experiments, a plasma is maintained in a steady state by con-tinuous ionization or injection of plasma to offset the losses. To calculatethe density profile in this case, we must add a source term to the equationof continuity:

dn- -D V*n = Q(r)dt

[5-351

The sign is chosen so that when Q is positive, it represents a sourceand contributes to positive an/at. In steady state, we set dn/dt = 0 andare left with a Poisson-type equation for n(r).

Constant Ionization Function

In many weakly ionized gases, ionization is produced by energetic elec-trons in the tail of the Maxwellian distribution. In this case, the sourceterm Q is proportional to the electron density 12. Setting Q = Zn, whereZ is the “ionization function,” we have

V*n = -(Z/D)n [5-361

165Diffusion and

Resistivity

5.3

5.3.1

This is the same equation as that for S, Eq. [5-25]. Consequently, thedensity profile is a cosine or Bessel function, as in the case of a decayingplasma, only in this case the density remains constant. The plasma ismaintained against diffusion losses by whatever heat source keeps theelectron temperature at its constant value and by a small influx of neutralatoms to replenish those that are ionized.

166ChapterFive

5.3.2 Plane Source

We next consider what profile would be obtained in slab geometry ifthere is a localized source on the plane x = 0. Such a source might be,for instance, a slit-collimated beam of ultraviolet light strong enough toionize the neutral gas. The steady state diffusion equation is then

d”n- -S(O)

djc2- D[5-571

Except at x = 0, the density must satisfy a*n/&* = 0. This obviously hasthe solution (Fig. 5-7)

I4n=no 1 - -( >L[5-581

The plasma has a linear profile. The discontinuity in slope at the sourceis characteristic of δ-function sources.

5.3.3 Line Source

Finally, we consider a cylindrical plasma with a source located on theaxis. Such a source might, for instance, be a beam of energetic electronsproducing ionization along the axis. Except at r = 0, the density mustsatisfy

la an- - r- =

( >o

r ar ar[5-391

The solution that vanishes at r = a is

n = no In (a/r) [5-401

- L 0 +L

FIGURE 5-7 The triangular density profileresulting from a plane sourceunder diffusion.

167Diffusion and

Resistivity

-r

a 0 aThe logarithmic density profile FIGURE 5-8resulting from a line source underdiffusion.

The density becomes infinite at r = 0 (Fig. 5-8); it is not possible todetermine the density near the axis accurately without considering thefinite width of the source.

RECOMBINATION 5.4

When an ion and an electron collide, particularly at low relative velocity,they have a finite probability of recombining into a neutral atom. Toconserve momentum, a third body must be present. If this third bodyis an emitted photon, the process is called radiative recombination. If it isa particle, the process is called three-body recombination. The loss of plasmaby recombination can be represented by a negative source term in theequation of continuity. It is clear that this term will be proportional tolz,rci = n*. In the absence of the diffusion terms, the equation of continuitythen becomes

dn/dt = -an2 [5-411

The constant of proportionality (Y is called the recombination coefficient and has units of m3/sec. Equation [5-41] is a nonlinear equation for 71.This means that the straightforward method for satisfying initial andboundary conditions by linear superposition of solutions is not available.Fortunately, Eq. [5.41] is such a simple nonlinear equation that the

168ChapterFive

solution can be found by inspection. It is

1 1

n (r,U=-+at

no(r)[5-441

where no(r) is the initial density distribution. It is easily verified that thissatisfies Eq. [5-41]. After the density has fallen far below its initial value,it decays reciprocally with time:

n oc 1/αt [5-481

This is a fundamentally different behavior from the case of diffusion,in which the time variation is exponential.

Figure 5-9 shows the results of measurements of the density decayin the afterglow of a weakly ionized H plasma. When the density is high,

I I I I I I 1 I I I I I I I I I I

O 1 2 3 4 5

t (msec)FIGURE 5-9 Density decay curves of a weakly ionized plasma under recombination and

diffusion. [From S. C. Brown, Basic Data of Plasma Physics, John Wiley and Sons,New York, 1959.]

169Diffusion and

Resistivity

DIFFUSION ACROSS A MAGNETIC FIELD 5.5

The rate of plasma loss by diffusion can be decreased by a magneticfield; this is the problem of confinement in controlled fusion research.Consider a weakly ionized plasma in a magnetic field (Fig. 5-10). Chargedparticles will move along B by diffusion and mobility according to Eq.[5-10], since B does not affect motion in the parallel direction. Thus wehave, for each species,

O 0 0 0 O O -II

A charged particle in a magnetic field will gyrate about FIGURE 5-10the same line of force until it makes a collision.

recombination, which is proportional to n 2, is dominant, and the density Idecays reciprocally. After the density has reached a low value, diffusion Ibecomes dominant, and the decay is thenceforth exponential.

If there were no collisions, particles would not diffuse at all in theperpendicular direction - they would continue to gyrate about the same-line of force. There are, of course, particle drifts across B because ofelectric fields or gradients in B, but these can be arranged to be parallelto the walls. For instance, in a perfectly symmetric cylinder (Fig. 5-11),the gradients are all in the radial direction, so that the guiding centerdrifts are in the azimuthal direction. The drifts would then be harmless.

When there are collisions, particles migrate across B to the wallsalong the gradients. They do this by a random-walk process (Fig. 5-12).When an ion, say, collides with a neutral atom, the ion leaves the collisiontraveling in a different direction. It continues to gyrate about the mag-netic field in the same direction, but its phase of gyration is changeddiscontinuously. (The Larmor radius may also change, but let us supposethat the ion does not gain or lose energy on the average.)

!

/

II

I

/I

170ChapterFive

FIGURE 5-11 Particle drifts in a cylindrically sym-metric plasma column do not lead tolosses.

FIGURE 5-12 Diffusion of gyrating par-ticles by collisions withneutral atoms.

The guiding center, therefore, shifts position in a collision and undergoesa random walk. The particles will diffuse in the direction opposite Vn.The step length in the random walk is no longer λm, as in magnetic-field-free diffusion, but has instead the magnitude of the Larmor radius rL.Diffusion across B can therefore be slowed down by decreasing rL; thatis, by increasing B.

To see how this comes about, we write the perpendicular componentof the fluid equation of motion for either species as follows:

dvlmndt

=bm(EfvlxB)-KTVn-mnvv=O r5-451

We have again assumed that the plasma is isothermal and that Y is largeenough for the dvl/dt term to be negligible. The x and y components are

171Diffusion and

Resistivity

mnvv, = *enE, - KT$ f envyB

anmnvw, = *enE, - KT- renv,B

9

[5-461

Using the definitions of p and D, we have

D dn O,74 =+E,----f-uY

[5-471D dn ocvy=fpEy---~-~,nay v

Substituting for vx, we may solve for u,:

D anv,(l+w:~~)=*pE,,----co:T~

EX 2 ,KTlan

n aygfoc7 eg-&g [5-481

where r = v-l. Similarly, vx is given by

D an E 2 ,KTlanv,(l+0272)=f~E,-nas+~~~2~r~c~ ---

eB n ay[5-49]

The last two terms of these equations contain the E X B and diamagneticdrifts:

EXUEy = --B

[5- 501

VDx= TKT 1 an-

vD~= *KTG!?l

eB n dy eB n ax

The first two terms can be simplified by defining the perpendicularmobility and diffusion coefficients:

P D, =D

pL= 1+w:T2 1 + U2T2c

With the help of Eqs. [5-50] and [5-51], we can write Eqs. [5-48] and[5-49] as

Vnvl=fpulE-DI--+ VE + VD

n 1+ (v’/co~)[5-521

172ChapterFive

From this, it is evident that the perpendicular velocity of eitherspecies is composed of two parts. First, there are usual VE and vo driftsperpendicular to the gradients in potential and density. These drifts areslowed down by collisions with neutrals; the drag factor 1 + (Y”/u~)becomes unity when v + 0. Second, there are the mobility and diffusiondrifts parallel to the gradients in potential and density. These drifts havethe same form as in the B = 0 case, but the coefficients p and D arereduced by the factor 1 + WIT*.

The product o,r is an important quantity in magnetic confinement.When wzr* << 1, the magnetic field has little effect on diffusion. Whenwzr* >> 1, the magnetic field significantly retards the rate of diffusionacross B. The following alternative forms for op can easily be verified: ~

WC7 = WC/V = /LB = A,,,/rL 15-531

In the limit ozr* >> 1, we haveKT 1 KTvD*=---=-mv a*~*c d

[5-541

Comparing with Eq. [5-8], we see that the role of the collision frequencyv has been reversed. In diffusion parallel to B, D is proportional to v-l,since collisions retard the motion. In diffusion perpendicular to B, DIis proportional to v, since collisions are needed for cross-field migration.The dependence on m has also been reversed. Keeping in mind that vis proportional to m-l’*, we see that D oc m-l’*, while DI oc m I’*. Inparallel diffusion, electrons move faster than ions because of their higherthermal velocity; in perpendicular diffusion, electrons escape more slowlybecause of their smaller Larmor radius.

Disregarding numerical factors of order unity, we may write Eq.[5-8] as

D = KT/mv -vKr -A$/7 [5-551

This form, the square of a length over a time, shows that diffusion is arandom-walk process with a step length λm. Equation [5-54] can be written

KTv 2

DA=--2

m2va,+v - -

vth 7[5-561

This shows that perpendicular diffusion is a random-walk process witha step length rL, rather than A,,,.

5.5.1 Ambipolar Diffusion across B

Because the diffusion and mobility coefficients are anisotropic in thepresence of a magnetic field, the problem of ambipolar diffusion is not

rilt

rel

173Diffusion and

Resistivity

Parallel and perpendicular particle fluxes in a magnetic field.

as straightforward as in the B = 0 case. Consider the particle fluxesperpendicular to B (Fig. 5-13). Ordinarily, since Fe1 is smaller than ril,a transverse electric field would be set up so as to aid electron diffusionand retard ion diffusion. However, this electric field can be short-circuitedby an imbalance of the fluxes along B. That is, the negative chargeresulting from rel < ril can be dissipated by electrons escaping alongthe field lines. Although the total diffusion must be ambipolar, theperpendicular part of the losses need not be ambipolar. The ions candiffuse out primarily radially, while the electrons diffuse out primarilyalong B. Whether or not this in fact happens depends on the particularexperiment. In short plasma columns with the field lines terminating onconducting plates, one would expect the ambipolar electric field to beshort-circuited out. Each species then diffuses radially at a different rate.In long, thin plasma columns terminated by insulating plates, one wouldexpect the radial diffusion to be ambipolar because escape along B isarduous.

-

Mathematically, the problem is to solve simultaneously the equationsof continuity [5-12] for ions and electrons. It is not the fluxes ri but thedivergences V * I’j which must be set equal to each other. SeparatingV . T’j into perpendicular and parallel components, we have

V * I’i = V, * (pilnEl_ Dil Vn) + $ pinE, - Di E

[5-573

?. r, = v, - (--pelnEl - D,.,.Vn) +$ -p,+E, - Deg

>

The equation resulting from settingV . I’i = V . r, cannot easily be separ-ated into one-dimensional equations. Furthermore, the answer dependssensitively on the boundary conditions at the ends of the field lines.Unless the plasma is so long that parallel diffusion can be neglectedaltogether, there is no simple answer to the problem of ambipolardiffusion across a magnetic field.

FIGURE 5-13

174ChapterFive

5.5.2 Experimental Checks

Whether or not a magnetic field reduces transverse diffusion in accord-ance with Eq. [5-51] became the subject of numerous investigations. Thefirst experiment performed in a tube long enough that diffusion to theends could be neglected was that of Lehnert and Hoh in Sweden. Theyused a helium positive column about 1 cm in diameter and 3.5 m long(Fig. 5-14). In such a plasma, the electrons are continuously lost by radialdiffusion to the walls and are replenished by ionization of the neutralgas by the electrons in the tail of the velocity distribution. These fastelectrons, in turn, are replenished by acceleration in the longitudinalelectric field. Consequently, one would expect E, to be roughly propor-tional to the rate of transverse diffusion. Two probes set in the wall ofthe discharge tube were used to measure E, as B was varied. The ratioof E,(B) to E,(O) is shown as a function of B in Fig. 5-15. At low B fields,the experimental points follow closely the predicted curve, calculated onthe basis of Eq. [5-52]. At a critical field B, of about 0.2 T, however, theexperimental points departed from theory and, in fact, showed an increaseof diffusion with B. The critical field B, increased with pressure, suggest-ing that a critical value of o,r was involved and that something wentwrong with the “classical” theory of diffusion when O,T was too large.

The trouble was soon found by Kadomtsev and Nedospasov in theU.S.S.R. These theorists discovered that an instability should develop athigh magnetic fields; that is, a plasma wave would be excited by the E,field, and that this wave would cause enhanced radial losses. The theorycorrectly predicted the value of Bc. The wave, in the form of a helicaldistortion of the plasma column, was later seen directly in an experimentby Allen, Paulikas, and Pyle at Berkeley. This helical instability of thepositive column was the first instance in which “anomalous diffusion”across magnetic fields was definitively explained, but the explanation was

CATHODEANODE

FIGURE 5-14 The Lehnert-Hoh experiment to check the effect of a magnetic field ondiffusion in a weakly ionized gas.

1.0

0.8

g 0.61miii

0.4

0.2

0

t+b\

∆\

%A AA

e \\\

1 %

?

??..

∆

0

175Diffusion and

Resistivity

0 0.2 0.4 0.6

B (T)

The normalized longitudinal electric field measured FIGURE 5-15as a function of B at two different pressures. Theoreti-cal curves are shown for comparison. [From F. C. Hohand B. Lehnert, Phys. Fluids 3, 600 (1960).]

applicable only to weakly ionized gases. In the fully ionized plasmas offusion research, anomalous diffusion proved to be a much tougherproblem to solve.

5-1. The electron-neutral collision cross section for 2-eV electrons in He is about PROBLEMS67ras, where a0 = 0.53 x lo-” cm is the radius of the first Bohr orbit of thehydrogen atom. A positive column with no magnetic field has p = 1 Torr ofHe (at room temperature) and KT, = 2 eV.

(a) Compute the electron diffusion coefficient in m2/sec, assuming that Cnraveraged over the velocity distribution is equal to o-v for 2-eV electrons.

(b) If the current density along column is 2 kA/m* and the plasma density islOI m-“, what is the electric field along the column?

176ChapterFive

5-2. A weakly ionized plasma slab in plane geometry has a density distribution

n(x) = 120 cos (42L) -LSxIL

The plasma decays by both diffusion and recombination. If L = 0.03 m, D =0.4 m’/sec, and α = lo-l5 mg/sec, at what density will the rate of loss by diffusionbe equal to the rate of loss by recombination?

5-3. A weakly ionized plasma is created in a cubical aluminum box of length Lon each side. It decays by ambipolar diffusion.

(a) Write an expression for the density distribution in the lowest diffusion mode.

(b) Define what you mean by the decay time constant and compute it if D, =1 O-’ m’/sec.

5-4. A long, cylindrical positive column has B = 0.2 T, KT< = 0.1 eV, and otherparameters the same as in Problem 5-1. The density profile is

n(r) = ~oJo(r/r~l”2)

with the boundary condition tr = 0 at r = a = 1 cm. Note: Jo(z) = 0 at z = 2.4.

(a) Show that the ambipolar diffusion coefficient to be used above can be approxi-mated by DA..

(b) Neglecting recombination and losses from the ends of the column, computethe confinement time r.

5-5. For the density profile of Fig. 5-7, derive an expression for the peak densityno in terms of the source strength Q and the other parameters of the problem.(Hint: E’quate the source per m2 to the particle flux to the walls per m2.)

5-6. You do a recombination experiment in a weakly ionized gas in which themain loss mechanism is recombination. You create a plasma of density IO*’ mm3by a sudden burst of ultraviolet radiation and observe that the density decaysto half its initial value in 10 msec. What is the value of the recombinationcoefficient ff ? Give units.

5.6 COLLISIONS IN FULLY IONIZED PLASMAS

When the plasma is composed of ions and electrons alone, all collisionsare Coulomb collisions between charged particles. However, there is adistinct difference between (a) collisions between like particles (ion-ionor electron-electron collisions) and (b) collisions between unlike particles(ion-electron or electron-ion collisions). Consider two identical particlescolliding (Fig. 5-16). If it is a head-on collision, the particles emerge withtheir velocities reversed; they simply interchange their orbits, and the

Shift of guiding centers of two like particles FIGURE 5-16 making a 90° collision.

two guiding centers remain in the same places. The result is the sameas in a glancing collision, in which the trajectories are hardly disturbed.The worst that can happen is a 90° collision, in which the velocities are changed 90° in direction. The orbits after collision will then be the dashed circles, and the guiding centers will have shifted. However, it is clearthat the “center of mass” of the two guiding centers remains stationary.For this reason, collisions between like particles give rise to very little diffusion.This situation is to be contrasted with the case of ions colliding withneutral atoms. In that case, the final velocity of the neutral is of noconcern, and the ion random-walks away from its initial position. In thecase of ion-ion collisions, however, there is a detailed balance in eachcollision; for each ion that moves outward, there is another that movesinward as a result of the collision.

When two particles of opposite charge collide, however, the situationis entirely different (Fig. 5-17). The worst case is now the 180° collision,in which the particles emerge with their velocities reversed. Since theymust continue to gyrate about the lines of force in the proper sense,both guiding centers will move in the same direction. Unlike-particlecollisions give rise to diffusion. The physical picture is somewhat differentfor ions and electrons because of the disparity in mass. The electronsbounce off the nearly stationary ions and random-walk in the usualfashion. The ions are slightly jostled in each collision and move aboutas a result of frequent bombardment by electrons. Nonetheless, becauseof the conservation of momentum in each collision, the rates of diffusionare the same for ions and electrons, as we shall show.

177Diffusion and

Resistivity

178ChapterFive

FIGURE 5-17 Shift of guiding centers of twooppositely charged particles mak-ing a 180° collision.

5.6.1 Plasma Resistivity

The fluid equations of motion including the effects of charged-particlecollisions may be written as follows (cf. Eq. [3-47]):

Mn dv.-=en(E+viXB)-Vpi-V.rri+Pi,dt

dvemnz

=-en(E+ve XB)-V@,-V*m,+Pei

[5- 581

The terms Pie and P,i represent, respectively, the momentum gain ofthe ion fluid caused by collisions with electrons, and vice versa. Thestress tensor Pj has been split into the isotropic part $j and the anisotropicviscosity tensor mj. Like-particle collisions, which give rise to stresseswithin each fluid individually, are contained in s+ Since these collisionsdo not give rise to much diffusion, we shall ignore the terms V . a+ Asfor the terms Pei and Pk, which represent the friction between the twofluids, the conservation of momentum requires

Pie = -P,i [5-591

We can write P,i in terms of the collision frequency in the usualmanner:

P,i = T?ZTZ(Vi - V,)V,i [5-601

and similarly for Pi,. Since the collisions are Coulomb collisions, onewould expect P,; to be proportional to the Coulomb force, which isproportional to e* (for singly charged ions). Furthermore, P,i must beproportional to the density of electrons 12, and to the density of scatteringcenters Ilip which, of course, is equal to n,. Finally, P,i should be propor-tional to the relative velocity of the two fluids. On physical grounds,then, we can write P,i as

P,i = Tje2?Z2(V; - V,) [5-611

where q is a constant of proportionality. Comparing this with Eq. [5-60],we see that

I Ine2I Iv,. = -qm

[5-621

I I

The constant q is the specific resistivity of the plasma; that this jibes withthe usual meaning of resistivity will become clear shortly.

Mechanics of Coulomb Collisions

When an electron collides with a neutral atom, no force is felt until theelectron is close to the atom on the scale of atomic dimensions; the

collisions are like billiard-ball collisions. When an electron collides withan ion, the electron is gradually deflected by the long-range Coulombfield of the ion. Nonetheless, one can derive an effective cross sectionfor this kind of collision. It will suffice for our purposes to give anorder-of-magnitude estimate of the cross section. In Fig. 5-18, an electronof velocity v approaches a fixed ion of charge e. In the absence of Coulombforces, the electron would have a distance of closest approach t-0, calledthe impact parameter. In the presence of a Coulomb attraction, the electronwill be deflected by an angle x, which is related to ro. The Coulomb forceis

F=- e2

z&7[5-631

179Diffusion and

Resistivity

5.6.2

180ChapterFive

\mv \

FIGURE 5-18 Orbit of an electron making a Coulomb collision with an ion.

This force is felt during the time the electron is in the vicinity of theion; this time is roughly

T = Q/V I5441

The change in the electron’s momentum is therefore approximately

A(mv) = lfll = ,,c3.00v [5-651

We wish to estimate the cross section for large-angle collisions, in whichx 190”. For a 90° collision, the change in mu is of the order of mu itself.Thus

∆(mv) = mu z e2/4~w-Ov, To = e2/4momv2

The cross section is then(T=&= e4/16~e~m2v4

The collision frequency is, therefore,

[5-661

[5-671

vei = m-v = ne4/16~e$m2vs [5-681

and the resistivity is

[5-691

For a Maxwellian distribution of electrons, we may replace v* by KT,/mfor our order-of-magnitude estimate:

2 l/2re m’ = (~~w,)~(KT~)~‘*

15-701

Equation [5-70] is the resistivity based on large-angle collisions alone.In practice, because of the long range of the Coulomb force, small-angle

collisions are much more frequent, and the cumulative effect of manysmall-angle deflections turns out to be larger than the effect of large-angle collisions. It was shown by Spitzer that Eq. [5-70] should be multipliedby a factor ln Λ:

7re2m If2

rl = (47r&3)2(KTe)3’2ln Λ 15.711

where

A = AD/?-O = 12 nn A; [5-721

This factor represents the maximum impact parameter, in units of ro asgiven by Eq. [5-66], averaged over a Maxwellian distribution. Themaximum impact parameter is taken to be λD because Debye shieldingsuppresses the Coulomb field at larger distances. Although Λ dependson n and RX’,, its logarithm is insensitive to the exact values of the plasmaparameters. Typical values of ln Λ are given below.

KT, (eV) 12 (m-‘) ln Λ

0.22

100lo4103

lOI10”lOI102’lo*’

9.1 (Q-machine)10.2 (lab plasma)13.7 (typical torus)16.0 (fusion reactor)6.8 (laser plasma)

It is evident that ln Λ varies only a factor of two as the plasma parametersrange over many orders of magnitude. For most purposes, it will besufficiently accurate to let ln Λ = 10 regardless of the type of plasmainvolved.

Physical Meaning of 17 5.6.3

Let us suppose that an electric field E exists in a plasma and that thecurrent that it drives is all carried by the electrons, which are much moremobile than the ions. Let B = 0 and KT, = 0, so that V . P, = 0. Then,in steady state, the electron equation of motion [5-58] reduces to

enE =Pei [5- 731

Diffusion andResistivity

182 Since j = en(vi - v,), Eq. [5-61] can be writtenChapterFive P& = Ten j [5-741

so that Eq. [5-73] becomes

E = qj [5-751

This is simply Ohm’s law, and the constant 17 is just the specific resistivity.The expression for r) in a plasma, as given by Eq. [5-71] or Eq. [5-69],has several features which should be pointed out.

(A) In Eq. [5-71], we see that q is independent of density (except forthe weak dependence in ln Λ). This is a rather surprising result, sinceit means that if a field E is applied to a plasma, the current j, as givenby Eq. [5-75], is independent of the number of charge carriers. Thereason is that although j increases with ne, the frictional drag against theions increases with ni. Since n, = ni, these two effects cancel. This cancella-tion can be seen in Eqs. [5-68] and [5-69]. The collision frequency v6i isindeed proportional to n, but the factor n cancels out in q. A fully ionizedplasma behaves quite differently from a weakly ionized one in this respect.In a weakly ionized plasma, we have j = -nev,, v, = -p,E, so that j =nep,E. Since p, depends only on the density of neutrals, the current isproportional to the plasma density n.

(B) Equation [5-71] shows that 77 is proportional to (KT,)-s’2. As aplasma is heated, the Coulomb cross section decreases, and the resistivitydrops rather rapidly with increasing temperature. Plasmas at thermonu-clear temperatures (tens of keV) are essentially collisionless; this is thereason so much theoretical research is done on collisionless plasmas. Ofcourse, there must always be some collisions; otherwise, there would notbe any fusion reactions either. An easy way to heat a plasma is simplyto pass a current through it. The I242 (or i2~) losses then turn up as anincrease in electron temperature. This is called ohmic heating. The(KT,)-3’2 dependence of r), however, does not allow this method to beused up to thermonuclear temperatures. The plasma becomes such agood conductor at temperatures above 1 keV that ohmic heating is avery slow process in that range.

(C) Equation [5-68] shows that vei varies as vm3. The fast electronsin the tail of the velocity distribution make very few collisions. Thecurrent is therefore carried mainly by these electrons rather than by thebulk of the electrons in the main body of the distribution. The strongdependence on v has another interesting consequence. If an electricfield is suddenly applied to a plasma, a phenomenon known as electronrunaway can occur. A few electrons which happen to be moving fast in

the direction of -E when the field is applied will have gained so much energy before encountering an ion that they can make only a glancing collision. This allows them to pick up more energy from the electric fieldand decrease their collision cross section even further. If E is large

enough, the cross section falls so fast that these runaway electrons nevermake a collision. They form an accelerated electron beam detached fromthe main body of the distribution.

Numerical Values of q 5.6.4

Exact computations of r) which take into account the ion recoil in each collision and are properly averaged over the electron distribution werefirst given by Spitzer. The following result for hydrogen is sometimes called the Spitzer resistivity:

qll = 5.2 x 1O-5Z ln Λ

T3/2ceVJ ohm-m [5-761

Here Z is the ion charge number, which we have taken to be 1 elsewherein this book. Since the dependence on M is weak, these values can alsobe used for other gases. The subscript I] means that this value of q is tobe used for motions parallel to B. For motions perpendicular to B, oneshould use nl given by

771 = 2.07711 [5-771

This does not mean that conductivity along B is only two times betterthan conductivity across B. A factor like wzr2 still has to be taken intoaccount. The factor 2.0 comes from a difference in weighting of thevarious velocities in the electron distribution. In perpendicular motions,the slow electrons, which have small Larmor radii, contribute more tothe resistivity than in parallel motions.

For KT, = 100 eV, Eq. [5-76] yields

n = 5 x lo-’ ohm-m

This is to be compared with various metallic conductors:

copper . . . . . . . . . . . . . . . n = 2 x 10m8 ohm-m

stainless steel . . . . . . . . . . n = 7 x lo-’ ohm-m

mercury . . . . . . . . . . . . . . q = 10M6 ohm-m

A 100-eV plasma, therefore, has a conductivity like that of stainless steel.

183Diffusion and

Resistivity

184ChapterFive

5.7 THE SINGLE-FLUID MHD EQUATIONS

We now come to the problem of diffusion in a fully ionized plasma.Since the dissipative term P,i contains the difference in velocities vi - v,,it is simpler to work with a linear combination of the ion and electronequations such that vi - v, is the unknown rather than vi or v, separately.Up to now, we have regarded a plasma as composed of two interpenetrat-ing fluids. The linear combination we are going to choose will describethe plasma as a single fluid, like liquid mercury, with a mass density pand an electrical conductivity 1 /.q. These are the equations of magnetohy-drodynamics (MHD).

For a quasineutral plasma with singly charged ions, we can define themass density p, mass velocity v, and current density j as follows:

p c7ZiM+?Z~?fZ zn(M+?TZ) [5-781

1v c -(niMVi + nemve) z

MVi + mv,

ρ M+m[5-791

j E e(niVi - ?Z,V,) * ?W(Vi - V,) [5-801

In the equation of motion, we shall add a term Mng for a gravitationalforce. This term can be used to represent any nonelectromagnetic forceapplied to the plasma. The ion and electron equations can be written

&$= en(E+vi XB)-Vfii +Mng+Pi, [5-811

mn$=-en(E+veXB)-Vfie+mng+Pei [5- 821

For simplicity, we have neglected the viscosity tensor 3, as we did earlier.This neglect does not incur much error if the Larmor radius is muchsmaller than the scale length over which the various quantities change.We have also neglected the (v . V)v terms because the derivation wouldbe unnecessarily complicated otherwise. This simplification is moredifficult to justify. To avoid a lengthy discussion, we shall simply say thatv is assumed to be so small that this quadratic term is negligible.

We now add Eqs. [5-81] and [5-82], obtaining

?Zi(MVi +??ZV,)=e?Z(V; -v,)XB-Vp+n(M+m)g [5-83]

The electric field has cancelled out, as have the collision terms P,i = ‘-Pk.We have introduced the notation

p = pi + PC [5-841

for the total pressure. With the help of Eqs. [5-78]-[5-80], Eq. [5-83] canbe written simply

p$=jXB-Vp+pg [5-851

This is the single-fluid equation of motion describing the mass flow. Theelectric field does not appear explicitly because the fluid is neutral. Thethree body forces on the right-hand side are exactly what one wouldhave expected.

A less obvious equation is obtained by taking a different linearcombination of the two-fluid equations. Let us multiply Eq. [5-81] by mand Eq. [5-82] by M and subtract the latter from the former. The resultis

+MVp,+M+m)P,i

With the help of Eqs. [5-78], [5-80], and [5-61], this becomes

Mmn 8 j-_ _( )dt n=epE-(M+m)ne77j-mVpi+MVP,

e

+ en(mvi + Mv,) X B

The last term can be simplified as follows:

mV;+MV,=MV;+mv,+M(V,-Vi)+m(Vi-V,)

.

=fv-(M-m)J

Dividing Eq. [5-87] by ep, we now have

[5-861

[5-871

[5-881

E+vXB-qj=$[Fi(i)+(M-m)jXB+mVpi-MVP,j

[5- 891

The a/at term can be neglected in slow motions, where inertial (i.e.,cyclotron frequency) effects are unimportant. In the limit m/M + 0, Eq.[5-89] then becomes

E+vXB=qj+i(jXB-VP,) [5- 901

185Diffusion and

Resistivity

186ChapterFive

This is our second equation, called the generalized Ohm’s law. It describesthe electrical properties of the conducting fluid. The j X B term is calledthe Hall current term. It often happens that this and the last term aresmall enough to be neglected; Ohm’s law is then simply

E+vXB=qj (5-911

Equations of continuity for mass p and charge u are easily obtainedfrom the sum and difference of the ion and electron equations ofcontinuity, The set of MHD equations is then as follows:

avp at-=jXB-V$+pg [5-851

E+vXB=vj 15.911

ap-&+vqpq=o

Together with Maxwell’s equations, this set is often used to describe theequilibrium state of the plasma. It can also be used to derive plasmawaves, but it is considerably less accurate than the two-fluid equationswe have been using. For problems involving resistivity, the simplicity ofthe MHD equations outweighs their disadvantages. The MHD equationshave been used extensively by astrophysicists working in cosmic electrody-namics, by hydrodynamicists working on MHD energy conversion, andby fusion theorists working with complicated magnetic geometries.

5.8 DIFFUSION IN FULLY IONIZED PLASMAS

In the absence of gravity, Eqs. [5-85] and [5-91] for a steady state plasmabecome

jXB=Vfi [5-941

E+vXB=vj [5-951

The parallel component of the latter equation is simply

which is the ordinary Ohm’s law. The perpendicular component is foundby taking the cross-product with B:

EXB+(v,XB)XB=qLjXB=qlV$

EXB-vlB2=qlVfi

ExB VIIVJ. =2-zVp

B[5-961

The first term is just the E x B drift of both species together. The secondterm is the diffusion velocity in the direction of -VP. For instance, in anaxisymmetric cylindrical plasma in which E and VP are in the radialdirection, we would have

187Diffusion and

Resistivity

ET qL afiv0=--B

v,=-p$

The flux associated with diffusion is

[5-971

This has the form of Fick’s law, Eq. [5-11], with the diffusion coefficient

DI=q*nCKT

B2 I [5-991

L I

This is the so-called “classical” diffusion coefficient for a fully ionized gas.Note that Dl is proportional to l/B’, just as in the case of weakly

ionized gases. This dependence is characteristic of classical diffusion andcan ultimately be traced back to the random-walk process with a steplength rL. Equation [5-99], however, differs from Eq. [5-54] for a partiallyionized gas in three essential ways. First, D, is not a constant in a fullyionized gas; it is proportional to n. This is because the density of scatteringcenters is not fixed by the neutral atom density but is the plasma densityitself. Second, since q is proportional to (KT)-3’2, DI decreases withincreasing temperature in a fully ionized gas. The opposite is true in apartially ionized gas. The reason for the difference is the velocity depen-dence of the Coulomb cross section. Third, diffusion is automaticallyambipolar in a fully ionized gas (as long as like-particle collisions areneglected). DL in Eq. [5-99] is the coefficient for the entire fluid; noambipolar electric field arises, because both species diffuse at the samerate. This is a consequence of the conservation of momentum in ion-

188ChapterFive

electron collisions. This point is somewhat clearer if ones uses the two-fluid equations (see Problem 5-15).

Finally, we wish to point out that there is no transverse mobility in afully ionized gas. Equation [5-96] for vL contains no component alongE which depends on E. If a transverse E field is applied to a uniformplasma, both species drift together with the E x B velocity. Since thereis no relative drift between the two species, they do not collide, and thereis no drift in the direction of E. Of course, there are collisions due tothermal motions, and this simple result is only an approximate one. Itcomes from our neglect of (a) like-particle collisions, (b) the electronmass, and (c) the last two terms in Ohm’s law, Eq. [5-90].

5.9 SOLUTKONS OF THE DIFFUSION EQUATION

Since DL is not a constant in a fully ionized gas, let us define a quantityA which is constant:

A = qKT/B2 [5- 1001

We have assumed that KT and B are uniform, and that the dependenceof q on n through the ln Λ factor can be ignored. For the case Ti = T,,we then have

DL = 2nA [5- 1011

The equation of continuity [5-921 can now be written

dn/dt=V*(D,Vn)=AV*(2nVn)

an/at = A V2n2 [5- 1021

This is a nonlinear equation for n, for which there are very few simplesolutions.

5.9.1 Time Dependence

If we separate the variables by ktting

n = T(t)S(r)

we can write Eq. [5-102] as

15.1031

where -1/τ is the separation constant. The spatial part of this equationis difficult to solve, but the temporal part is the same equation that weencountered in recombination, Eq. [5-41]. The solution, therefore, is

.J5- 1041

189Diffusion and

Resistivity

At large times t, the density decays as 1/t, as in the case of recombination.This reciprocal decay is what would be expected of a fully ionized plasmadiffusing classically. The exponential decay of a weakly ionized gas is adistinctly different behavior.

Time-Independent Solutions 5.9.2

There is one case in which the diffusion equation can be solved simply.Imagine a long plasma column (Fig. 5-19) with a source on the axis whichmaintains a steady state as plasma is lost by radial diffusion and recombi-nation. The density profile outside the source region will be determinedby the competition between diffusion and recombination. The densityfalloff distance will be short if diffusion is small and recombination islarge, and will be long in the opposite case. In the region outside thesource, the equation of continuity is

-A VYn2 = -an2 [5- 105]

This equation is linear in n2 and can easily be solved. In cylindricalgeometry, the solution is a Bessel function. In plane geometry, Eq. [5- 105]reads

a%' a----a2ax2 -A

15.1061

t

B ->

Diffusion of a fully ionized cylindrical plasma across a magnetic field. FIGURE 5-19

190ChapterFive

with the solution

n* = n$ exp [-(a/A)“* x] [5- 1071

The scale distance is

1 = (A/a)“* [5- 108]

Since A changes with magnetic field while a remains constant, the changeof l with B constitutes a check of classical diffusion. This experimentwas actually tried on a Q-machine, which provides a fully ionized plasma.Unfortunately, the presence of asymmetric E x B drifts leading toanother type of loss-by convection-made the experiment inconclusive.

Finally, we wish to point out a scaling law which is applicable to anyfully ionized steady state plasma maintained by a constant source Q ina uniform B field. The equation of continuity then reads

-A V2n2 = -qKT V2(n2/B2) = Q [5- 1091

Since n and B occur only in the combination n/B, the density profilewill remain unchanged as B is changed, and the density itself will increaselinearly with B :

nKB [5-1101

One might have expected the equilibrium density n to scale as B*, sinceDI oc B-*; but one must remember that D, is itself proportional to n.

5.10 BOHM DIFFUSION AND NEOCLASSICAL DIFFUSION

Although the theory of diffusion via Coulomb collisions had been knownfor a long time, laboratory verification of the l/B* dependence of DIin a fully ionized plasma eluded all experimenters until the 1960s. Inalmost all previous experiments, DI scaled as B-l, rather than B-*, andthe decay of plasmas was found to be exponential, rather than reciprocal,with time. Furthermore, the absolute value of DI was far larger thanthat given by Eq. [5-99]. This anomalously poor magnetic confinementwas first noted in 1946 by Bohm, Burhop, and Massey, who weredeveloping a magnetic arc for use in uranium isotope separation. Bohmgave the semiempirical formula

DL=--’ KT’ED16f?B B

[5-1111

This formula was obeyed in a surprising number of different experi-ments. Diffusion following this law is called Bohm diffusion. Since n,, isindependent of density, the decay is exponential with time. The timeconstant in a cylindrical column of radius R and length L can be estimatedas follows:

N rmR”L nRr = dNldt = I’,27rRL

=-2I’,

where N is the total number of ion-electron pairs in the plasma. Withthe flux T, given by Fick’s law and Bohm’s formula, we have

nR nR R2’ z 2DB an/& 7 2DBn/R = 20, s ‘I3

15-1121

The quantity Tag is often called the Bohm time.Perhaps the most extensive series of experiments verifying the Bohm

formula was done on a half-dozen devices called stellarators at Princeton.A stellarator is a toroidal magnetic container with the lines of forcetwisted so as to average out the grad-B and curvature drifts describedin Section 2.3. Figure 5-20 shows a compilation of data taken over adecade on many different types of discharges in the Model C Stellarator.The measured values of r lie near a line representing the Bohm timera. Close adherence to Bohm diffusion would have serious consequencesfor the controlled fusion program. Equation [5-111] shows that D13increases, rather than decreases, with temperature, and though itdecreases with B, it decreases more slowly than expected. In absolutemagnitude, De is also much larger than D,. For instance, for a 100-eVplasma in 1-T field, we have

D = 1 (102)(1.6x lo-“‘)* 16 (1.6x 10-“)(l)

= 6.25 m’/sec

If the density is 1019 rnp3, the classical diffusion coefficient is

DI

= 2nKTrllB2

= (2)( lO’“)( 102)( 1.6 x lo-‘“)

m2

x (2.0)(5.2 x 10-5)(10)(loo)“‘*

= (320) (1.04 x 10-6) = 3.33 X lo-’ m’/sec

The disagreement is four orders of magnitude.Several explanations have been proposed for Bohm diffusion. First,

there is the possibility of magnetic field errors. In the complicated

191Diffusion and

Resistiity

192ChapterFive

-0.1

\0 ALKALI PLASMA

\(DATA NORMALIZED TO12.3 KG, 5.0 cm RADIUS)

ELECTRON CYCLOTRONHEATING

RESISTIVE MICROWAVEHEATING

LOHMIC HEATING: \AFTERGLOW

BOHM DIFFUSION .

OHMICHEATING

ION CYCLOTRONRESONANCE HEATING

0.1 1.0 10/0,/B (arb. units)

100

FIGURE 5-20 Summary of confinement time measurements taken on various types of dis-charges in the Model C Stellarator, showing adherence to the Bohm diffusionlaw. [Courtesy of D. J. Grove, Princeton University Plasma Physics Laboratory,sponsored by the U.S. Atomic Energy Commission.]

geometries used in fusion research, it is not always clear that the linesof force either close upon themselves or even stay within the chamber.Since the mean free paths are so long, only a slight asymmetry in themagnetic coil structure will enable electrons to wander out to the wallswithout making collisions. The ambipolar electric field will then pull theions out. Second, there is the possibility of asymmetric electric fields.These can arise from obstacles inserted into the plasma, from asym-metries in the vacuum chamber, or from asymmetries in the way theplasma is created or heated. The dc E x B drifts then need not be parallelto the walls, and ions and electrons can be carried together to the wallsby E x B convection. The drift patterns, called convective cells, have beenobserved. Finally, there is the possibility of oscillating electric fields arising

from unstable plasma waves. If these fluctuating fields are random, theE x B drifts constitute a collisionless random-walk process. Even if the oscillating field is a pure sine wave, it can lead to enhanced losses becausethe phase of the E x B drift can be such that the drift is always outwardwhenever the fluctuation in density is positive. One may regard thissituation as a moving convective cell pattern. Fluctuating electric fieldsare often observed when there is anomalous diffusion, but in many cases,it can be shown that the fields are not responsible for all of the losses.All three anomalous loss mechanisms may be present at the same timein experiments on fully ionized plasmas.

The scaling of Dg with KT, and B can easily be shown to be thenatural one whenever the losses are caused by E x B drifts, either station-ary or oscillating. Let the ‘escape flux be proportional to the E x B driftvelocity:

rl = nvI a nE/B [5- 1131

Because of Debye shielding, the maximum potential in the plasma isgiven by

&nax = KT, [5- 1141

If R is a characteristic scale length of the plasma (of the order of itsradius), the maximum electric field is then

E4 max KT

*ax z---R eR

This leads to a flux I’* given by

Vn = -DB Vn

[5-1151

[5-1161

where y is some fraction less than unity. Thus the fact that Dg isproportional to KT,/eB is no surprise. The value γ = & has no theoreticaljustification but is an empirical number agreeing with most experimentsto within a factor of two or three.

Recent experiments on toroidal devices have achieved confinementtimes of order 100~. This was accomplished by carefully eliminatingoscillations and asymmetries. However, in toroidal devices, other effectsoccur which enhance collisional diffusion. Figure 5-21 shows a torus withhelical lines of force. The twist is needed to eliminate the unidirectionalgrad-B and curvature drifts. As a particle follows a line of force, it seesa larger 1 Bj near the inside wall of the torus and a smaller ) BI near theoutside wall. Some particles are trapped by the magnetic mirror effect

193Diffusion and

Resistivity

194ChapterFive

INCREASING

FIGURE 5-21 A banana orbit of a particle confined in the twisted magnetic field of a toroidal confinement device. The “orbit” is really the locus of pointsat which the particle crosses the plane of the paper.

JBANANA DIFFUSION

MODIFIED! PLATEAU ; CLASSICAL

V

FIGURE 5-22 Behavior of the neoclassical diffusion coefficient withcollision frequency v.

and do not circulate all the way around the torus. The guiding centersof these trapped particles trace out banana-shaped orbits as they makesuccessive passes through a given cross section (Fig. 5-21). As a particlemakes collisions, it becomes trapped and untrapped successively andgoes from one banana orbit to another. The random-walk step lengthis therefore the width of the banana orbit rather than rL, and the“classical” diffusion coefficient is increased. This is called neoclassicaldiffusion. The dependence of DI on Y is shown in Fig. 5-22. In theregion of small Y, banana diffusion is larger than classical diffusion. Inthe region of large Y, there is classical diffusion, but it is modified by

5-7. Show that the mean free path Ari for electron-ion collisions is proportional PROBLEMSto T:.

5-8. A Tokamak is a toroidal plasma container in which a current is driven inthe fully ionized plasma by an electric field applied along B (Fig. P5-8). Howmany V/m must be applied to drive a total current of 200 kA in a plasma withh.T, = 500 eV and a cross-sectional area of 75 cm2?

5-9. Suppose the plasma in a fusion reactor is in the shape of a cylinder 1.2 min diameter and 100 m long. The 5-T magnetic field is uniform except for shortmirror regions at the ends, which we may neglect. Other parameters are KTi =20 keV, KT, = 10 keV, and n = 102’ me3 (at r = 0). The density profile is foundexperimentally to be approximately as sketched in Fig. P5-9.

(a) Assuming classical diffusion, calculate D, at r = 0.5 m.

(b) Calculate dN/dt, the total number of ion-electron pairs leaving the centralregion radially per second.

FIGURE P5-8

r (cm) FIGURE P5-9

currents along B. The theoretical curve for neoclassical diffusion has been observed experimentally by Ohkawa at La Jolla, California.

195Diffusion and

Resistivity

196 (c) Estimate the confinement time r by T = -fV/(c.W/dl). Note: a rough estimateChapterFive

is all that can be expected in this type of problem. The profile has obviouslybeen affected by processes other than classical diffusion.-

5-10. Estimate the classical. diffusion time of a plasma cylinder 10 cm in radius,with n = !O*’ m-‘, KT. = KTi = 10 keV, B = 5 T.

5-11. A cylindrical plasma column has a density distribution

n = ~(1 - ?/a’)

where a = 10 cm and no = 10” m-s. If KT. = 100 eV, KTi = 0, and the axialmagnetic field B. is 1 T, what is the ratio between the Bohm and the classicaldiffusion coefficients perpendicular to Bo?

5-12. A weakly ionized plasma can still be governed by Spitzer resistivity ifv,i >> v,,), where v,o is the electron-neutral collision frequency. Here are somedata for the electron-neutral momentum transfer cross section ueo in squareangstroms (A’):

E = 2eV E = 10 eV

Helium 6.3 4.1Argon 2.5 13.8

For singly ionized He and A plasmas with KT, = 2 and 10 eV (4 cases), estimatethe fractional ionizationf = ni/(no + ni) at which v,i = veo, assuming that the valueof i%(T,) can be crudely approximated by a(E)m(E), where E = KT.. (Hint:For v,,,, use Eq. [7-11]; for v,i, use Eqs. [5-62] and [5-76].

5-13. The plasma in a toroidal stellarator is ohmically heated by a current alongB of lo5 A/mP. The density is uniform at n = 10” mm3 and does not change.The Joule heat T$ goes to the electrons. Calculate the rate of increase of KT. ineV/psec at the time when KT, = 10 eV.

5-14. In a e-pinch, a large current is discharged through a one-turn coil. Therising magnetic field inside the coil induces a surface current in the highlyconducting plasma. The surface current is opposite in direction to the coil currentand hence keeps the magnetic field out of the plasma. The magnetic field pressurebetween the coil and the plasma then compresses the plasma. This can workonly if the magnetic field does not penetrate into the plasma during the pulse.Using the Spitzer resistivity, estimate the maximum pulse length for a hydrogend-pinch whose initial conditions are KT, = 10 eV, n = IO** m-‘, r = 2 cm, if thefield is to penetrate only 1/10 of the way to the axis.

5-15. Consider an axisymmetric cylindrical plasma with E = E,i, B = Bi, andVpi = Vfi, = i@/ar. If we neglect the (v * V)v term, which is tantamount to neglect-ing the centrifugal force, the steady state two-fluid equations can be written inthe form

en(E+vi XB)-Vpi-e*n*rl(vi-~.)=O

-en(E+v.XB)-Vp,+e2n2q(vi-~.)=0

FIGURE P5- 14

(a) From the .4 components of these equations, show that Vi, = vu,.

(b) From the r components, show that vi0 = vE + voj (j = i, e).

(c) Find an expression for vi, showing that it does not depend on Er.

5-16. Use the single-fluid MHD equation of motion and the mass continuityequation to calculate the phase velocity of an ion acoustic wave in an unmagnet-ized, uniform plasma with T, >> Ti.

5-17 Calculate the resistive damping of Alfven waves by deriving the dispersionrelation from the single-fluid equations [5-85] and [5-91] and Maxwell’s equations[4-72] and [4-77]. Linearize and neglect gravity, displacement current, and VP.

(a) Show that

(b) Find an explicit expression for Im (k) when o is real and 1) is small.

5-18. If a cylindrical plasma diffuses at the Bohm rate, calculate the steady stateradial density profile n(r), ignoring the fact that it may be unstable. Assume thatthe density is zero at r = 00 and has a value n, at r = rO.

5-19. A cylindrical column of plasma in a uniform magnetic field B = B,i carriesa uniform current density j = j& where i is a unit vector parallel to the axis ofthe cylinder.

(a) Calculate the magnetic field B(r) produced by this plasma current.

(b) Write an expression for the grad-B drift of a charged particle with VII= 0 interms of B,, jZ, r, vI, q, and m. You may assume that the field calculated in (a)is small compared to B, (but not zero).

(c) If the plasma has electrical resistivity, there is also an electric field E = E,;.Calculate the azimuthal electron drift due to this field, taking into account thehelicity of the B field.

(d) Draw a diagram showing the direction of the drifts in (b) and (c) for bothions and electrons in the (r, 0) plane.